In [1]}, the choice \(\alpha =\frac{7 - 2\gamma }{6} \)  was made to maximize the stability region of the resulting scheme, but this coefficient is indeed a free parameter
and can also be chosen in different ways, as long as stability is not compromised. In order to identify possible alternative choices, we perform an analysis using the concepts introduced in [2]}, [3]}, [4]}
(see also the review in [5]}). A similar analysis for the implicit part of the IMEX scheme was carried out in [6]}, to which we refer for a summary of the related theoretical results.
We then define
\(A = \begin{bmatrix}0 & 0 & 0 \\\gamma & 0 & 0 \\1-\alpha & \alpha & 0\end{bmatrix} \qquad b^T = \begin{bmatrix}\frac{1}{2}-\frac{\gamma }{4} & \frac{1}{2}-\frac{\gamma }{4} & \frac{\gamma }{2}\end{bmatrix}\) 
